DOCSLIB.ORG

Lightweight Linear Types in System F°

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lightweight Linear Types in System F° University of Pennsylvania ScholarlyCommons Departmental Papers (CIS) Department of Computer & Information Science 1-23-2010 Lightweight linear types in System F° Karl Mazurak University of Pennsylvania Jianzhou Zhao University of Pennsylvania Stephan A. Zdancewic University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/cis_papers Part of the Computer Sciences Commons Recommended Citation Karl Mazurak, Jianzhou Zhao, and Stephan A. Zdancewic, "Lightweight linear types in System F°", . January 2010. Karl Mazurak, Jianzhou Zhao, and Steve Zdancewic. Lightweight linear types in System Fo. In ACM SIGPLAN International Workshop on Types in Languages Design and Implementation (TLDI), pages 77-88, 2010. doi>10.1145/1708016.1708027 © ACM, 2010. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in ACM SIGPLAN International Workshop on Types in Languages Design and Implementation, {(2010)} http://doi.acm.org/10.1145/1708016.1708027" Email [email protected] This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_papers/591 For more information, please contact [email protected]. Lightweight linear types in System F° Abstract We present Fo, an extension of System F that uses kinds to distinguish between linear and unrestricted types, simplifying the use of linearity for general-purpose programming. We demonstrate through examples how Fo can elegantly express many useful protocols, and we prove that any protocol representable as a DFA can be encoded as an Fo type. We supply mechanized proofs of Fo's soundness and parametricity properties, along with a nonstandard operational semantics that formalizes common intuitions about linearity and aids in reasoning about protocols. We compare Fo to other linear systems, noting that the simplicity of our kind-based approach leads to a more explicit account of what linearity is meant to capture, allowing otherwise-conflicting interpretations of linearity (in particular, restrictions on aliasing versus restrictions on resource usage) to coexist peacefully. We also discuss extensions to Fo aimed at making the core language more practical, including the additive fragment of linear logic, algebraic datatypes, and recursion. Disciplines Computer Sciences Comments Karl Mazurak, Jianzhou Zhao, and Steve Zdancewic. Lightweight linear types in System Fo. In ACM SIGPLAN International Workshop on Types in Languages Design and Implementation (TLDI), pages 77-88, 2010. doi>10.1145/1708016.1708027 © ACM, 2010. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in ACM SIGPLAN International Workshop on Types in Languages Design and Implementation, {(2010)} http://doi.acm.org/ 10.1145/1708016.1708027" Email [email protected] This other is available at ScholarlyCommons: https://repository.upenn.edu/cis_papers/591 Lightweight Linear Types in System F◦ Karl Mazurak Jianzhou Zhao Steve Zdancewic University of Pennsylvania fmazurak,jianzhou,[email protected] Abstract Given these success stories, it is perhaps surprising that we have ◦ yet to see general linear types seriously considered for inclusion in We present System F , an extension of System F that uses kinds to 1 distinguish between linear and unrestricted types, simplifying the a mainstream functional programming language. But alas, linear use of linearity for general-purpose programming. We demonstrate types can easily lead to awkward programming models and poten- through examples how System F◦ can elegantly express many use- tially complicated language designs that are difficult both to imple- ment and to program with. This paper seeks to address these issues ful protocols, and we prove that any protocol representable as a ◦ DFA can be encoded as an F◦ type. We supply mechanized proofs by introducing System F —pronounced “F-pop”—a language that ◦ is intended to be a simple foundation for practical linear program- of System F ’s soundness and parametricity properties, along with ◦ a nonstandard operational semantics that formalizes common intu- ming. Rather than aiming at one particular problem, System F lets itions about linearity and aids in reasoning about protocols. programmers enforce their own protocol abstractions through the We compare System F◦ to other linear systems, noting that the power of linearity and polymorphism, yet its typing discipline is lightweight enough to expose in a surface language. simplicity of our kind-based approach leads to a more explicit ac- ◦ count of what linearity is meant to capture, allowing otherwise- System F is simply the Girard–Reynolds polymorphic λ- conflicting interpretations of linearity (in particular, restrictions on calculus [14, 23] extended with two base kinds: ?, classifying aliasing versus restrictions on resource usage) to coexist peace- ordinary, unrestricted types, and ◦, classifying linear types. A sub- fully. We also discuss extensions to System F◦ aimed at making kinding relation ? ≤ ◦ makes explicit the observation that values the core language more practical, including the additive fragment of unrestricted types may safely be treated linearly; i.e., since vari- of linear logic, algebraic datatypes, and recursion. ables of unrestricted type may be used any number of times and linear variables must be used exactly once, it is always safe to use Categories and Subject Descriptors D.3.3 [Programming Lan- unrestricted variables as though they were linear. Any System F◦ guages]: Language Constructs and Features expression with kinds erased is a well-typed System F expression. In introducing System F◦, this paper contributes the following: General Terms Design, Languages, Theory • The design of System F◦, and in particular its use of kinds and Keywords Linear logic, Polymorphism, Type systems kind subsumption, which is lightweight and structured so as to integrate well with existing functional languages. (Section 2) 1. Introduction • Mechanized proofs of standard soundness and parametricity re- sults for System F◦, which ensure that the properties functional Linear logic [15, 16] models resource usage by restricting the prop- programmers rely on continue to hold. (Section 2.1) erties of contraction (the ability to duplicate a resource) and weak- ◦ ening (the ability to discard a resource). In the context of program- • A second, linearity-aware semantics for System F , which for- ming languages, ideas from linear logic were quickly adopted, at malizes common intuitions about linearity and shows that these first to eliminate garbage collection [21] and shortly thereafter to intuitions do indeed hold. (Section 4) handle mutable state [29]. • Several examples—including all regular languages—along Since their introduction, variants, refinements, and improve- with extensions and proposals for compiler support, remark- ments on linear type systems have been proposed for many ap- able primarily for their simplicity, which showcase System F◦’s plications, including explicit memory management and control of potential usefulness. (Sections 3 and 5) aliasing [2, 13, 17, 28, 36], capabilities [9, 10], and tracking state In the rest of this section we discuss the design of System F◦ in changes for program analysis [11, 32]. Of particular interest is work ◦ on typestates, which ensure that a sequence of API calls is well- the context of prior work, and we showcase System F ’s ability to behaved [13, 12], and on session types, which check that the end- enforce programmer-defined protocols with a familiar example. points of a channel agree on the next message to be sent or re- 1.1 Prior work: Linear type system design considerations ceived [18, 25, 27]. Walker has a more comprehensive survey [33]. There are many variants on linear type systems in the litera- ture [33], but the crucial design decision from our perspective is how linear and unrestricted variables are differentiated and how that mechanism interacts with polymorphism. Our use of kinds Permission to make digital or hard copies of all or part of this work for personal or and kind subsumption is intended to capture the essence of linear- classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation ity simply and generally while remaining faithful to the standard on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. 1 Clean is often seen as the exception to this, but there are subtle differences TLDI’10 January 23, 2010, Madrid, Spain. between its uniqueness types, which concern aliasing, and standard linear Copyright c 2010 ACM 978-1-60558-891-9/10/01. $5.00 types (see Section 3.1). semantics and programming model of System F. This division be- become overwhelming if exposed to the programmer. As we are tween linear and unrestricted types is inspired by that proposed for incorporating the full power of System F, exposing at least some monomorphic systems by Wadler [29] and Benton [5]. Here we types is unavoidable; we also believe that System F◦ convincingly contrast our approach with some alternatives from the literature. demonstrates that just the simple distinction between linear and un- Broadly speaking, there are two other approaches to distinguish-
Load more

Recommended publications
  • Proving Termination of Evaluation for System F with Control Operators
    Proving termination of evaluation for System F with control operators Małgorzata Biernacka Dariusz Biernacki Sergue¨ıLenglet Institute of Computer Science Institute of Computer Science LORIA University of Wrocław University of Wrocław Universit´ede Lorraine [email protected] [email protected] [email protected] Marek Materzok Institute of Computer Science University of Wrocław [email protected] We present new proofs of termination of evaluation in reduction semantics (i.e., a small-step opera- tional semantics with explicit representation of evaluation contexts) for System F with control oper- ators. We introduce a modified version of Girard’s proof method based on reducibility candidates, where the reducibility predicates are defined on values and on evaluation contexts as prescribed by the reduction semantics format. We address both abortive control operators (callcc) and delimited- control operators (shift and reset) for which we introduce novel polymorphic type systems, and we consider both the call-by-value and call-by-name evaluation strategies. 1 Introduction Termination of reductions is one of the crucial properties of typed λ-calculi. When considering a λ- calculus as a deterministic programming language, one is usually interested in termination of reductions according to a given evaluation strategy, such as call by value or call by name, rather than in more general normalization properties. A convenient format to specify such strategies is reduction semantics, i.e., a form of operational semantics with explicit representation of evaluation (reduction) contexts [14], where the evaluation contexts represent continuations [10]. Reduction semantics is particularly convenient for expressing non-local control effects and has been most successfully used to express the semantics of control operators such as callcc [14], or shift and reset [5].
    [Show full text]
  • Lecture Notes on Types for Part II of the Computer Science Tripos
    Q Lecture Notes on Types for Part II of the Computer Science Tripos Prof. Andrew M. Pitts University of Cambridge Computer Laboratory c 2016 A. M. Pitts Contents Learning Guide i 1 Introduction 1 2 ML Polymorphism 6 2.1 Mini-ML type system . 6 2.2 Examples of type inference, by hand . 14 2.3 Principal type schemes . 16 2.4 A type inference algorithm . 18 3 Polymorphic Reference Types 25 3.1 The problem . 25 3.2 Restoring type soundness . 30 4 Polymorphic Lambda Calculus 33 4.1 From type schemes to polymorphic types . 33 4.2 The Polymorphic Lambda Calculus (PLC) type system . 37 4.3 PLC type inference . 42 4.4 Datatypes in PLC . 43 4.5 Existential types . 50 5 Dependent Types 53 5.1 Dependent functions . 53 5.2 Pure Type Systems . 57 5.3 System Fw .............................................. 63 6 Propositions as Types 67 6.1 Intuitionistic logics . 67 6.2 Curry-Howard correspondence . 69 6.3 Calculus of Constructions, lC ................................... 73 6.4 Inductive types . 76 7 Further Topics 81 References 84 Learning Guide These notes and slides are designed to accompany 12 lectures on type systems for Part II of the Cambridge University Computer Science Tripos. The course builds on the techniques intro- duced in the Part IB course on Semantics of Programming Languages for specifying type systems for programming languages and reasoning about their properties. The emphasis here is on type systems for functional languages and their connection to constructive logic. We pay par- ticular attention to the notion of parametric polymorphism (also known as generics), both because it has proven useful in practice and because its theory is quite subtle.
    [Show full text]
  • The System F of Variable Types, Fifteen Years Later
    Theoretical Computer Science 45 (1986) 159-192 159 North-Holland THE SYSTEM F OF VARIABLE TYPES, FIFTEEN YEARS LATER Jean-Yves GIRARD Equipe de Logique Mathdmatique, UA 753 du CNRS, 75251 Paris Cedex 05, France Communicated by M. Nivat Received December 1985 Revised March 1986 Abstract. The semantic study of system F stumbles on the problem of variable types for which there was no convincing interpretation; we develop here a semantics based on the category-theoretic idea of direct limit, so that the behaviour of a variable type on any domain is determined by its behaviour on finite ones, thus getting rid of the circularity of variable types. To do so, one has first to simplify somehow the extant semantic ideas, replacing Scott domains by the simpler and more finitary qualitative domains. The interpretation obtained is extremely compact, as shown on simple examples. The paper also contains the definitions of a very small 'universal model' of lambda-calculus, and investigates the concept totality. Contents Introduction ................................... 159 1. Qualitative domains and A-structures ........................ 162 2. Semantics of variable types ............................ 168 3. The system F .................................. 174 3.1. The semantics of F: Discussion ........................ 177 3.2. Case of irrt ................................. 182 4. The intrinsic model of A-calculus .......................... 183 4.1. Discussion about t* ............................. 183 4.2. Final remarks ...............................
    [Show full text]
  • An Introduction to Logical Relations Proving Program Properties Using Logical Relations
    An Introduction to Logical Relations Proving Program Properties Using Logical Relations Lau Skorstengaard [email protected] Contents 1 Introduction 2 1.1 Simply Typed Lambda Calculus (STLC) . .2 1.2 Logical Relations . .3 1.3 Categories of Logical Relations . .5 2 Normalization of the Simply Typed Lambda Calculus 5 2.1 Strong Normalization of STLC . .5 2.2 Exercises . 10 3 Type Safety for STLC 11 3.1 Type safety - the classical treatment . 11 3.2 Type safety - using logical predicate . 12 3.3 Exercises . 15 4 Universal Types and Relational Substitutions 15 4.1 System F (STLC with universal types) . 16 4.2 Contextual Equivalence . 19 4.3 A Logical Relation for System F . 20 4.4 Exercises . 28 5 Existential types 29 6 Recursive Types and Step Indexing 34 6.1 A motivating introduction to recursive types . 34 6.2 Simply typed lambda calculus extended with µ ............ 36 6.3 Step-indexing, logical relations for recursive types . 37 6.4 Exercises . 41 1 1 Introduction The term logical relations stems from Gordon Plotkin’s memorandum Lambda- definability and logical relations written in 1973. However, the spirit of the proof method can be traced back to Wiliam W. Tait who used it to show strong nor- malization of System T in 1967. Names are a curious thing. When I say “chair”, you immediately get a picture of a chair in your head. If I say “table”, then you picture a table. The reason you do this is because we denote a chair by “chair” and a table by “table”, but we might as well have said “giraffe” for chair and “Buddha” for table.
    [Show full text]
  • CS 6110 S18 Lecture 32 Dependent Types 1 Typing Lists with Lengths
    CS 6110 S18 Lecture 32 Dependent Types When we added kinding last time, part of the motivation was to complement the functions from types to terms that we had in System F with functions from types to types. Of course, all of these languages have plain functions from terms to terms. So it's natural to wonder what would happen if we added functions from values to types. The result is a language with \compile-time" types that can depend on \run-time" values. While this arrangement might seem paradoxical, it is a very powerful way to express the correctness of programs; and via the propositions-as-types principle, it also serves as the foundation for a modern crop of interactive theorem provers. The language feature is called dependent types (i.e., types depend on terms). Prominent dependently-typed languages include Coq, Nuprl, Agda, Lean, F*, and Idris. Some of these languages, like Coq and Nuprl, are more oriented toward proving things using propositions as types, and others, like F* and Idris, are more oriented toward writing \normal programs" with strong correctness guarantees. 1 Typing Lists with Lengths Dependent types can help avoid out-of-bounds errors by encoding the lengths of arrays as part of their type. Consider a plain recursive IList type that represents a list of any length. Using type operators, we might use a general type List that can be instantiated as List int, so its kind would be type ) type. But let's use a fixed element type for now. With dependent types, however, we can make IList a type constructor that takes a natural number as an argument, so IList n is a list of length n.
    [Show full text]
  • System F and Existential Types 15-312: Foundations of Programming Languages
    Recitation 6: System F and Existential Types 15-312: Foundations of Programming Languages Serena Wang, Charles Yuan February 21, 2018 We saw how to use inductive and coinductive types last recitation. We can also add polymorphic types to our type system, which leads us to System F. With polymorphic types, we can have functions that work the same regardless of the types of some parts of the expression.1 1 System F Inductive and coinductive types expand the expressivity of T considerably. The power of type operators allows us to genericly manipulate data of heterogeneous types, building new types from old ones. But to write truly \generic" programs, we want truly polymorphic expressions| functions that operate on containers of some arbitrary type, for example. To gain this power, we add parametric polymorphism to the language, which results in System F, introduced by Girand (1972) and Reynolds (1974). Typ τ ::= t type variable τ1 ! τ2 function 8(t.τ) universal type Exp e ::= x variable λ (x : τ) e abstraction e1(e2) application Λ(t) e type abstraction e[τ] type application Take stock of what we've added since last time, and what we've removed. The familiar type variables are now baked into the language, along with the universal type. We also have a new form of lambda expression, one that works over type variables rather than expression variables. What's missing? Nearly every other construct we've come to know and love! As will be the case repeatedly in the course, our tools such as products, sums, and inductive types are subsumed by the new polymorphic types.
    [Show full text]
  • 1 Polymorphism in Ocaml 2 Type Inference
    CS 4110 – Programming Languages and Logics .. Lecture #25: Type Inference . 1 Polymorphism in OCaml In languages lik OCaml, programmers don’t have to annotate their programs with 8X: τ or e [τ]. Both are automatically inferred by the compiler, although the programmer can specify types explicitly if desired. For example, we can write let double f x = f (f x) and Ocaml will gure out that the type is ('a ! 'a) ! 'a ! 'a which is roughly equivalent to 8A: (A ! A) ! A ! A We can also write double (fun x ! x+1) 7 and Ocaml will infer that the polymorphic function double is instantiated on the type int. The polymorphism in ML is not, however, exactly like the polymorphism in System F. ML restricts what types a type variable may be instantiated with. Specically, type variables can not be instantiated with polymorphic types. Also, polymorphic types are not allowed to appear on the left-hand side of arrows—i.e., a polymorphic type cannot be the type of a function argument. This form of polymorphism is known as let-polymorphism (due to the special role played by let in ML), or prenex polymorphism. These restrictions ensure that type inference is decidable. An example of a term that is typable in System F but not typable in ML is the self-application expression λx. x x. Try typing fun x ! x x in the top-level loop of Ocaml, and see what happens... 2 Type Inference In the simply-typed lambda calculus, we explicitly annotate the type of function arguments: λx:τ: e.
    [Show full text]
  • From System F to Typed Assembly Language
    From System F to Typed Assembly Language GREG MORRISETT and DAVID WALKER Cornell University KARL CRARY Carnegie Mellon University and NEAL GLEW Cornell University We motivate the design of a typed assembly language (TAL) and present a type-preserving transla- tion from System F to TAL. The typed assembly language we present is based on a conventional RISC assembly language, but its static type system provides support for enforcing high-level language abstractions, such as closures, tuples, and user-defined abstract data types. The type system ensures that well-typed programs cannot violate these abstractions. In addition, the typ- ing constructs admit many low-level compiler optimizations. Our translation to TAL is specified as a sequence of type-preserving transformations, including CPS and closure conversion phases; type-correct source programs are mapped to type-correct assembly language. A key contribution is an approach to polymorphic closure conversion that is considerably simpler than previous work. The compiler and typed assembly language provide a fully automatic way to produce certified code, suitable for use in systems where untrusted and potentially malicious code must be checked for safety before execution. Categories and Subject Descriptors: D.2.11 [Software Engineering]: Software Architectures— Languages; D.3.1 [Programming Languages]: Formal Definitions and Theory—Semantics, Syntax; D.3.2 [Programming Languages]: Language Classifications—Macro and Assembly Languages; D.3.4 [Programming Languages]: Processors—Compilers; F.3.2
    [Show full text]
  • Polymorphism All the Way Up! from System F to the Calculus of Constructions
    Programming = proving? The Curry-Howard correspondence today Second lecture Polymorphism all the way up! From System F to the Calculus of Constructions Xavier Leroy College` de France 2018-11-21 Curry-Howard in 1970 An isomorphism between simply-typed λ-calculus and intuitionistic logic that connects types and propositions; terms and proofs; reductions and cut elimination. This second lecture shows how: This correspondence extends to more expressive type systems and to more powerful logics. This correspondence inspired formal systems that are simultaneously a logic and a programming language (Martin-Lof¨ type theory, Calculus of Constructions, Pure Type Systems). 2 I Polymorphism and second-order logic Static typing vs. genericity Static typing with simple types (as in simply-typed λ-calculus but also as in Algol, Pascal, etc) sometimes forces us to duplicate code. Example A sorting algorithm applies to any list list(t) of elements of type t, provided it also receives then function t ! t ! bool that compares two elements of type t. With simple types, to sort lists of integers and lists of strings, we need two functions with dierent types, even if they implement the same algorithm: sort list int :(int ! int ! bool) ! list(int) ! list(int) sort list string :(string ! string ! bool) ! list(string) ! list(string) 4 Static typing vs. genericity There is a tension between static typing on the one hand and reusable implementations of generic algorithms on the other hand. Some languages elect to weaken static typing, e.g. by introducing a universal type “any” or “?” with run-time type checking, or even by turning typing o: void qsort(void * base, size_t nmemb, size_t size, int (*compar)(const void *, const void *)); Instead, polymorphic typing extends the algebra of types and the typing rules so as to give a precise type to a generic function.
    [Show full text]
  • Type Theory & Functional Programming
    Type Theory & Functional Programming Simon Thompson Computing Laboratory, University of Kent March 1999 c Simon Thompson, 1999 Not to be reproduced i ii To my parents Preface Constructive Type theory has been a topic of research interest to computer scientists, mathematicians, logicians and philosophers for a number of years. For computer scientists it provides a framework which brings together logic and programming languages in a most elegant and fertile way: program development and verification can proceed within a single system. Viewed in a different way, type theory is a functional programming language with some novel features, such as the totality of all its functions, its expressive type system allowing functions whose result type depends upon the value of its input, and sophisticated modules and abstract types whose interfaces can contain logical assertions as well as signature information. A third point of view emphasizes that programs (or functions) can be extracted from proofs in the logic. Up until now most of the material on type theory has only appeared in proceedings of conferences and in research papers, so it seems appropriate to try to set down the current state of development in a form accessible to interested final-year undergraduates, graduate students, research workers and teachers in computer science and related fields – hence this book. The book can be thought of as giving both a first and a second course in type theory. We begin with introductory material on logic and functional programming, and follow this by presenting the system of type theory itself, together with many examples. As well as this we go further, looking at the system from a mathematical perspective, thus elucidating a number of its important properties.
    [Show full text]
  • Type Theory: Impredicative Part (1/5)
    Type Theory: Impredicative Part (1/5) An Introduction to System F Alexandre Miquel Paris-sud University – Orsay, France [email protected] Introduction • System F : independently discovered by Girard (1970) and Reynolds (1974) • Quite different motivations... Girard: Interpretation of second-order logic Reynolds: Functional programming ... connected by the Curry-Howard isomorphism • Significant influence on the development of Type Theory – Interpretation of higher-order logic [Girard, Martin-L¨of] – Type:Type [Martin-L¨of1971] – Martin-L¨ofType Theory [1972, 1984, 1990, ..] – The Calculus of Constructions [Coquand 1984] 1 System F : syntax Types A, B ::= α | A → B | ∀α . B Terms t, u ::= x | λx : A . t | tu | Λα . t | tA | {z } | {z } term abstr./app. type abstr./app. FV (t) ≡ set of free (term) variables of the term t TV (t) ≡ set of free type variables of the term t TV (A) ≡ set of free type variables of the type A t{x := u} ≡ substitute the term u to the variable x in the term t t{α := A} ≡ substitute the type A to the type variable α in the term t B{α := A} ≡ substitute the type A to the type variable α in the type B Perform α-conversions to avoid capture of free (term/type) variables! 2 System F : Typing rules Contexts Γ, ∆ ::= x1 : A1, . , xn : An – Declarations are unordered – Declaration of type variables is implicit (for each α ∈ TV (Γ)) – Could also declare type variables explicitely: α : ∗. (just a matter of taste) Typing rules are syntax-directed: (x:A)∈Γ Γ ` x : A Γ; x : A ` t : B Γ ` t : A → B Γ ` u : A Γ ` λx : A .
    [Show full text]
  • Notes: Barendregt's Cube and Programming with Dependent Types
    Notes: Barendregt's cube and programming with dependent types Eric Lu I. ANNOTATED BIBLIOGRAPHY A. Lambda Calculi with Types @incollection{barendregt:1992 author = {Barendregt, Henk}, title = {Lambda Calculi with Types}, booktitle = {Handbook of Logic in Computer Science}, publisher = {Oxford University Press}, year = {1992}, editor = {Abramsky, S. and Gabbay, D.M. and Maibaum, T.S.E.}, volume = {2}, pages = {117--309}, address = {New York, New York} } Summary: As setup for introducing the lambda cube, Barendregt introduces the type-free lambda calculus and reviews some of its properties. Subsequently he presents accounts of typed lambda calculi from Curry and Church, including some Curry-style systems not belonging to the lambda cube (λU and λµ). He then describes the lambda cube construction that was first noted by Barendregt in 1991. The lambda cube describes an inclusion relation amongst eight typed lambda calculi. Moreover it explains a fine structure for the calculus of constructions arising from the presence or absence of three axiomatic additions to the simply-typed lambda calculus. The axes of the lambda cube are intuitively justified by the notion of dependency. After introducing the lambda cube, Barendregt gives properties of the languages belonging to the lambda cube, as well as some properties of pure type systems, which generalize the means of description of the axioms of the systems in the lambda cube. Evaluation: This work originates as a reference from the logical/mathematical community, but its content was important in providing a firm foundation for types in programming languages. The lambda cube explains prior work on a number of type systems by observing a taxonomy and relating them to logical systems via the Curry-Howard isomorphism.
    [Show full text]